Back to the school of divisibility!

\ \ \ \ \ \ We all have learned about divisibility tests for many certain numbers in our elementary school. In this note, I will be discussing about the method that those tests are derived from. For every derivation that I will be discussing, we will be taking the variable number of the form

I will just present you with those special cases that you are familiar with and you can apply this idea to create divisibility test for any positive integer you may want to. Also I will be using the term 'number' only and only for the set of positive integers or natural numbers in this context. Also keep in mind that you must have at least some basic knowledge of modular arithmetic to understand the derivations.

Let's begin!

Divisibility by 2‾\large \underline{\text{Divisibility by 2}}Divisibility by 2​

Statement

Any number with 2,4,6,82, \ 4, \ 6, \ 82,4,6,8 and 000 as the unit digit is divisible by 222.

Since, a0a_0a0​ is a single digit number. The only values that satisfy are 0,2,4,60, \ 2, \ 4, \ 60,2,4,6 and 888.

The same approach can be used for 555 and 101010 as well due to the fact that 10k10^k10k where k≥1k \ge 1k≥1 is always divisible by 555 and 101010 as well and hence, the values fitting for a0a_0a0​ in this case are 000 and 555 for the number 555 and 000 for the number 101010, thus proving the divisibility tests of 555 and 101010.

Thus, the sum of digits must be divisible by 333 for the number to be divisible by 333.

The same approach can be used for 999 as well due to the fact that 10k−110^k - 110k−1 where k≥1k \ge 1k≥1 is always divisible by 999 as well and hence, the sum of digits of the number in this case must be divisible by 999 so that the number itself is divisible by 999, thus proving the divisibility test of 999.

Thus, the if the tens and units place of a number taken in that order is divisible by 444 then the number is also divisible by 444.

The same approach can be used for 252525 as well due to the fact that 10k10^k10k where k≥2k \ge 2k≥2 is always divisible by 252525 as well and hence, if the digits in the tens and units place of a number taken in that order respectively, if divisible by 252525, then the number is also divisible by 252525.

Thus, the if the hundreds, tens and units place of a number taken in that order is divisible by 888 then the number is also divisible by 888.

The same approach can be used for 125125125 as well due to the fact that 10k10^k10k where k≥3k \ge 3k≥3 is always divisible by 125125125 as well and hence, if the digits in the hundreds, tens and units place of a number taken in that order respectively, if divisible by 125125125, then the number is also divisible by 125125125.

Divisibility by 11‾\large \underline{\text{Divisibility by 11}}Divisibility by 11​

Statement

Any number whose absolute difference of the sum of digits occurring in the even positions and the sum of digits occurring in odd positions, if it is 000 or divisible by 111111, then the number is also divisible by 111111.

From the above two conditions, we infer that for a number to be divisible by 111111, its absolute difference of the sum of digits occurring in the even positions and the sum of digits occurring in odd positions, should be 000 or divisible by 111111.

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@Harsh Shrivastava
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That is a weak part of our system. If a student is keen to learn more about a certain concept, their teachers must put their best efforts to help them go through these concepts. Moreover, it is the fact that our teachers have the burden of keeping things straight to syllabus and that's part of the problem that even they don't think it is important for them to help a child go through the concepts that are beyond the school syllabus.

Thanks for encouraging me to write this wiki. You can check it out here. Feel free to share your opinion about the wiki and it would be very kind of you if you can contribute by citing problems and examples.

I agree that adding the proofs for these divisibility rules will be helpful for those who want to learn about the underlying number theoretic. It would be great if you can contribute to Proof Of Divisibility Rules and we can link it from there.

@Calvin Lin
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Thanks for encouraging me to write this wiki. You can check it out here. Feel free to share your opinion about the wiki and it would be very kind of you if you can contribute by citing problems and examples.